Problem: Let $f(x)=-x^7+7x^6$. For what values of $x$ does the graph of $f$ have a point of inflection? Choose all answers that apply: Choose all answers that apply: (Choice A) A $x=0$ (Choice B) B $x=5$ (Choice C) C $x=7$ (Choice D) D $f$ has no points of inflection.
Solution: We can find the inflection points of the graph of $f$ by looking for the intervals where its second derivative $f''$ is positive/negative. This analysis is very similar to finding minimum/maximum points, only instead of analyzing $f'$, we are analyzing $f''$. The second derivative of $f$ is $f''(x)=-42x^4(x-5)$. $f''(x)=0$ for $x=0, 5$. Since $f''$ is a polynomial, it's defined for all real numbers. Therefore, our possible inflection points are $x=0$ and $x=5$. Our possible inflection points divide the number line into three intervals: $\llap{-}3$ $\llap{-}2$ $\llap{-}1$ $0$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $(-\infty,0)$ $(0,5)$ $(5,\infty)$ Let's evaluate $f''$ at each interval to see if it's positive or negative on that interval. Interval $x$ -value $f''(x)$ Verdict $(-\infty,0)$ $x=-1$ $f''(-1)=252>0$ $f$ is concave up $\cup$ $(0,5)$ $x=1$ $f''(1)=168>0$ $f$ is concave up $\cup$ $(5,\infty)$ $x=6$ $f''(6)=-54432<0$ $f$ is concave down $\cap$ We can see that the graph of $f$ changes concavity at $x=5$. In conclusion, $f$ has a point of inflection at $x=5$.